One can regard the possibility of digital compression as a failure of sensor design. If it is possible to compress measured data, one might argue that too many measurements were taken. - David Brady Compressive sensing main idea is to measure and compress to cope with the scarcity of resources. For example the limited resource can be battery power or limited communication band-width in simple sensors or measurement time in magnetic-resonance imaging. This talk first explains what it means to measure in a compressed mode and then how we can use prior knowledge about the structure of the signals to measure them in a compressed mode. Finally it shows how the prior knowledge used in compressive sensing moved forward over the few years from simple sparsity constraints to more advanced assumptions about the manifold of measurements. Such manifolds can be constructed in a data-driven manner for example by deep-learning. The talk briefly touches upon the statistical physics of inference and uses compressive sensing as a case study where information and computational complexity meet.